Place potatoes in a large pot, cover with salted water, and bring to a boil. Reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain; steam-dry for 1 to 2 minutes.

Preheat the oven to 375 degrees F (190 degrees C).

Add ¼ cup butter, ¼ cup milk, salt, and pepper to potatoes; mash until combined. Stir in cheese.

Fit one crust in a 9-inch pie plate; fill with potato mixture. Place remaining pie crust on filling, fold crust edges together, and crimp with a fork to seal.

Combine 2 teaspoons melted butter and 2 teaspoons milk in a small bowl; brush over crust. Cut several vent holes in top crust.

Bake in the preheated oven until crust is golden brown, about 45 minutes; if edges are getting too brown, cover with aluminum foil.